Learning Objective

At the end of this section, the learner will be able to define measures of variability, distinguish between range, variance, and standard deviation, and interpret standard deviation values in relation to the normal distribution.

Range

The simplest measure of variability is the range, calculated as the difference between the highest and lowest values in the dataset.
Although useful, the range is unstable because it depends solely on two extreme values and can change easily with outliers.

Standard Deviation (SD)

A more reliable and informative measure of dispersion is the standard deviation (SD). It reflects how far individual values typically lie from the mean.

Calculation steps:

1. Subtract the mean from each observation to obtain deviations (positive and negative values).
2. Square each deviation so all values become positive.
3. Add the squared deviations and divide by the number of observations to obtain the average of squared deviations.
4. Take the square root of this value — that result is the standard deviation.

Mathematically:

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The variance (s²) is the square of the standard deviation.

Interpretation in Normal Distributions

In a normal (bell-shaped) distribution, the standard deviation has predictable proportions:

• Within 1 SD of the mean: ~ 68% of observations

• Within 2 SDs of the mean: ~ 95.5% of observations

• Within 3 SDs of the mean: ~ 99.7% of observations

These percentages are foundational for understanding probability, confidence levels, and interpretation of statistical findings.